Finite Group DatabasebetaFinite Group Databasebeta| order | 13140 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(Dic365⋊C3) |
| unambiguous semidirect product | Dic1095⋊C3 |
| derived | C3×C5×C73 ⋊ C3×C4 |
| Hall | C3×C5×C73⋊C3 ⋊ C4 |
| Hall | C3×C73⋊C3 ⋊ Dic5 |
| Hall | C5×C73 ⋊ C3×Dic3 |
| Sylow | C73 ⋊ C3×Dic15 |
| Sylow | C5 ⋊ C3⋊(Dic73⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊(C5⋊(D73⋊C3)) |
| commutator, hypocenter | C3×C5×C73 | C3×C4 |
| Fitting, socle, cosocle | C2×C3×C5×C73 | C2×C3 |
| prime radical | C2×C3×C5×C73⋊C3 | C2 |
| LCS | G ▷ C3×C5×C73 | |
| UCS | C1 ◁ C2 | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 4380 |
A minimal faithful permutation representation — 2 generators on 85 points:
(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,41)(12,26,22,20,16,13)(19,46,47,30,31,60)(21,44,45,33,29,38)(23,25,43,35,36,39)(32,48,57,34,49,63)(37,42,62,50,51,58)(40,67,65,54,55,53)(52,61,68,56,64,72)(59,73,71,69,70,66)(75,76)(77,78)(80,81)(82,83,84,85);(1,2,5,11,19,32,33,20,4,8,13,23,37,52,54,36,10,16,24,38,53,66,69,51,18,29,39,41,57,68,73,64,31,48,55,58,60,47,62,67,49,46,61,70,72,63,28,43,44,27,42,59,71,65,45,15,26,14,25,40,56,50,35,22,7,6,12,21,34,30,17,9,3)(74,75,77,78,76)(79,80,81)Polycyclic presentation:
2,3^2,5,73,2|a^2=f,c^a=c^2,d^a=d^4,e^a=e^72,e^b=e^8414 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13140, 63);